bioRxiv preprint first posted online Mar. 24, 2016; doi: http://dx.doi.org/10.1101/045591. The copyright holder for this preprint (which was
not peer-reviewed) is the author/funder. All rights reserved. No reuse allowed without permission.
Relative contributions of conformational selection and induced fit
Denis Michel
Universite de Rennes1-IRSET. Campus Santé de Villejean. 35000 Rennes France. [email protected]
ABSTRACT. A long standing debate in biochemistry is to determine whether the conformational
changes observed during biomolecular interactions proceed through conformational selection (of preexisting
isoforms) or induced fit (ligand-induced 3D reshaping).
The latter mechanism had been invoked in certain circumstances, for example to explain the non-Michaelian
activity of monomeric enzymes like glucokinase. But the
relative importance of induced fit has been recently depreciated in favor of conformational selection, assumed
to be always sufficient, predominant in general and in
particular for glucokinase. The relative contributions of
conformational selection and induced fit are reconsidered
here in and out of equilibrium, in the light of earlier concepts such as the cyclic equilibrium rule and the turning
wheel of Wyman, using single molecule state probability, one way fluxes and net fluxes. The conditions for
a switch from conformational selection to induced fit at
a given ligand concentration are explicitly determined.
Out of equilibrium, the inspection of the enzyme states
circuit shows that conformational selection alone would
give a Michaelian reaction rate but not the established
nonlinear behavior of glucokinase. Moreover, when induced fit and conformational selection coexist and allow
kinetic cooperativity, the net flux emerging in the linkage
cycle necessarily corresponds to the induced fit path.
have been identified, which differ by the presence of a
unique or several binding sites on the macromolecule.
Multisite cooperativity, frequently encountered for regulatory enzymes, can proceed through either conformational selection or induced fit, illustrated by the MWC
[9] and KNF [10] models respectively. A general specificity of multisite cooperativity is to hold as well in
equilibrium and nonequilibrium conditions, as long evidenced by the oxygenation of vertebrate hemoglobin that
is not an enzyme. By contrast, single site cooperativity is more rarely observed and exists out of equilibrium only. These two modes of cooperativity have been
pertinently called cooperativity through space (between
the different sites at a given time point) and through
time (between the successive states of the same site) [11].
The latter model, of kinetic cooperativity, has been developed for the enzyme glucokinase/ATP-D-glucose 6phosphotransferase EC 2.7.1.1-hexokinase D, enriched in
the liver and pancreas of vertebrates. Glucokinase displays a non-Michaelian reaction rate with respect to its
substrate (glucose), that is not observed for its counterparts catalyzing the same reaction in other tissues. The
mechanism of kinetic cooperativity has been clearly described by Rabin [12] and then modeled and supported
experimentally for glucokinase [13, 14, 15, 16]. Its principle can be visualized intuitively using the scheme of
Fig.1.
Keywords:Conformational selection; induced fit; kinetic
cooperativity; glucokinase; steady state.
Folding and binding are intertwined processes involved in intra- and inter-molecular interactions. It is
generally accepted that macromolecular interactions are
rarely rigid (lock and key) and involve conformational
changes. A persistent question in this context is whether
these changes result from the selection of a subset of preexisting conformations (conformational selection: CS)
or from a post-binding stereo-adjustment (induced fit:
IF). [1, 2, 3, 4, 5, 6, 7, 8]. CS and IF are important
phenomena which can stabilize specific folding states
and underly nonlinear biochemical phenomena such as
sigmoidal reactions. Sigmoidal or cooperative saturation curves are themselves essential for regulating biochemical systems and their possible multistability. Two
main modes of sigmoidal saturation by a single ligand
Figure 1. The nonlinear dependence on glucose of the glucokinase activity can be explained by a competition between
relaxation of E ∗ into E and the direct rebinding of the substrate to E ∗ . At low concentration of glucose (triangles),
transconformation takes place, repriming the long cycle with
its inefficient initial association, whereas at higher concentration of glucose, a molecule binds immediately to the enzyme.
In this short cycle, the enzyme remains in its reactive form
so that the global reaction rate is speeded up.
1
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where b (time−1 ), r (time−1 ), a (concentration−1
time−1 ) and d (time−1 ) are the rate constants of bending
of E into E ∗ , reversion to the basal conformation, association and dissociation respectively. As shown below,
this scheme always gives a Michaelian reaction rate.
In its basal conformation (E), the enzyme binds the
substrate with a low affinity, but once bound, the substrate induces the transconformation of the enzyme into
a new form E ∗ that is ”kinetically favourable but thermodynamically unfavourable” [12], that is to say less stable
than E ∗ in absence of substrate, but conveniently folded
for catalyzing the reaction. After release of the reaction
product (oval in Fig.1), a kinetic competition begins between two events: the relaxation of the enzyme to its
basal conformation and the reattachment of a new glucose. For a fixed relaxation rate constant, the issue of
this race depend on glucose concentration. The physiological outcome of this mechanism is remarkable as it
greatly contributes to the regulation of blood glucose levels. All the hexokinases catalyse the transformation of
glucose into glucose 6-phosphate (G6P), but the fate of
G6P depends on the organs. While it is consumed in
most tissues, the liver has the capacity to temporarily
store it in the form of glycogen, but only if the concentration of glucose in the blood exceeds a certain threshold. This exquisite mechanism is based on an induced-fit
mechanism. However, using the criterion of the relaxation times [2], authors revised the importance of IF and
suggested that CS is in fact predominant in general [3, 5]
and in particular for glucokinase [2, 17], for which different models have been proposed, such as those compiled in [18] and earlier ones [16]. In the alternative
approach proposed here, CS and IF are tested alone or
in combination for their capacity to yield the well established non-Michaelian activity of glucokinase. While
relaxation times are obtained from a transient return
to pure IF and CS equilibrium (Appendix A) [2, 19],
enzymatic reaction rates are measured in driven nonequilibrium steady states. Besides, the predominance of
CS over IF described in [3] concerns multimeric cooperativity in equilibrium (for hemoglobin) or under the quasiequilibrium hypothesis (for enzymes like aspartate transcarbamylase [9]), but the cooperativity of monomeric
enzymes like glucokinase can not be obtained in this way
[20]. In this study, the substrates will be considered as
rigid small molecules. We will adopt the classical approximation, as old as the theory of Michaelis-Menten,
that the substrates are much more numerous than the
macromolecules, which allows using pseudo-first order
constants of association.
1
1.1
1.1.1
The relative concentration of the different forms of the
enzyme are simply linked by equilibrium constants Kb =
b/r and Ka = a/d. The reaction rate is proportional to
the fraction of enzyme in the form E ∗ S
a
1.1.2
(1)
In nonequilibrium steady state
When c is not negligible, we obtain the familiar hyperbolic reaction rate of Briggs and Haldane
VM [S]
KM + [S]
with the maximal velocity
v=
VM = c
(2a)
(2b)
and
(b + r)(c + d)
(2c)
ab
CS has been shown predominant for glucokinase [2], but
we see here that it is clearly incapable alone to explain
kinetic cooperativity. Let us now examine the pure induced fit mechanism.
KM =
1.2
The pure induced fit mechanism is
necessarily cyclical
At first glance, the scheme
a
b
c
−
−
*
E+S −
)
−*
− ES −
)
−
− E*S −−→ E* + P
r
d
seems symmetrical to the CS scheme by permuting
the conformational transition and the binding reaction;
but in fact the above scheme is clearly incomplete because the enzyme should recover its basal conformation
to bind a new substrate in case of pure IF. An additional
backward transition (r∗ ) is thus necessary to connect the
final form E ∗ and the initial form E, thereby giving a cycle. Like the CS scheme, this minimal IF cycle is merely
Michaelian, with new expressions for VM and KM .
traditional
cbr∗
bc + (b + c + r)r∗
(3a)
(bc + cd + dr)r∗
a(bc + (b + c + r)r∗ )
(3b)
VM =
and
The reaction following pure CS is
b
Ka Kb [S]
1 + Kb + Ka Kb [S]
v=c
Pure CS and IF enzymatic reactions
Pure CS gives
Michaelian velocity
Under the quasi-equilibrium approximation
KM =
a
c
−
*
−
*
E−
)
−
− E* + S −
)
−
− E*S −−→ E*+P
r
d
2
bioRxiv preprint first posted online Mar. 24, 2016; doi: http://dx.doi.org/10.1101/045591. The copyright holder for this preprint (which was
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In the pure IF scheme, the isoform E ∗ does not accommodate directly a new substrate molecule, but there
is no obvious reason that a substrate can not bind to E ∗
before it relaxes. This would give the full CS/IF cycle
described below.
and in the same manner
kIF = a1 [S]
b2
d1 + b2
(4b)
kCS and kIF are hyperbolic and linear functions of
[S] respectively. Obviously these functions can intersect
if the initial slope of the hyperbola (b1 a2 /r1 ) is higher
2 The IF/CS dual cycle
than the slope of kIF when the relaxation rate constant
is low enough such that
2.1 The IF/CS dual cycle at equilibrium
a2
d1
r1
The complete cycle of enzyme state transitions is repre<
1+
(5)
b1
a1
b2
sented in Fig.2. This cycle is more realistic than pure CS
or IF because even if certain enzyme states have a very
In this case, there is a certain value of [S] below which
low probability, they are nonetheless not forbidden.
kCS > kIF and over which kCS < kIF . The critical substrate concentration is
d1
r1
b1
1+
−
(6)
[S] =
a1
b2
a2
The above global rates of CS and IF for a single
molecule E are always valid, but it is also interesting
to consider the number of molecules to which they apply.
The product of single molecule rates by the concentration
of the molecules involved, is called a flux. Expectedly,
the one-way flux approach of [1] gives the same result
(Appendix B). Moreover, when defining the global rates
of reverse CS (k−CS ) and reverse IF (k−IF ) (Fig.2), it
can be easily shown (Appendix C) that the forward and
reverse fluxes are strictly identical
Figure 2. First-order binding scheme coupling CS and IF.
kCS , k−CS , kIF and k−IF are the global rate constants of
CS, reverse CS, IF and reverse IF respectively.
2.1.1
The CS-IF switch
a2 [S]
r1 + a2 [S]
(7a)
kIF [E] = k−IF [E ∗ S]
(7b)
and
The relative importance of CS vs IF in equilibrium has
been evaluated through different ways, including the result of relaxation experiments [2] and one-way fluxes [1].
However, this predominance may be not absolute if it
can pass from one mechanism to another depending on
the conditions. Precisely, authors proposed that a higher
ligand concentration could favor IF [1, 21], as supported
by simulations, provided the conformational transition
is slow enough [22]. The dependence on the substrate of
the relative contribution of CS and IF can be explicitly
predicted by single molecule probability. This method
is based on the comparative probability that a single
molecule E reaches the state E ∗ S via either E ∗ (CS,
with a global rate constant kCS ) or via ES (IF, with a
global rate constant kIF ) (Fig.2). The global rate constants kCS and kIF are the reciprocal of the mean times
of first arrival to E ∗ S and can be understood as ”conditional rates” as follows. In the example of the CS path,
the global rate is the rate that E commits first to E ∗
and that once at this state, it continues forward until
E ∗ S instead of reverting back to E. The probability of
this event is a2 [S]/(r1 + a2 [S]) [23], which gives
kCS = b1
kCS [E] = k−CS [E ∗ S]
The possible predominance of a path, either CS or IF,
is the same for complex formation and dismantlement,
owing to microscopic reversibility.
2.1.2
The CS-IF detailed balance
The generalized reversibility of forward and backward
fluxes at equilibrium is known as the detailed balance
and holds for any transition, as minor as it can be, in
a network of any size [24]. As a consequence for the
cyclic network of Fig.2, the 8 rate constants are mutually constrained by the detailed balance rule (Appendix
D) reading
a1 b2 d2 r1 = a2 b1 d1 r2
(8a)
or differently written with the equilibrium constants
Ka1
Kb1
=
Ka2
Kb2
(8b)
All studies aimed at determining the relative im(4a) portance of CS and IF at equilibrium (for non-enzyme
3
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macromolecules or in case of enzymes, using nonmetabolizable substrates or a single substrate for a twosubstrate enzyme), are plausible only if the values of the
constants satisfy Eq.(8). This criterion is for example
verified for the rate constants given for Flavodoxin in [1],
but strangely not in all the models developed for glucokinase. At equilibrium, the clockwise (IF) and counterclockwise (CS) fluxes of Fig.2 cancel each other and
the saturation curve is an hyperbolic function of the substrate:
Y =
with
Kdapp =
[S]
+ [S]
(9a)
1 + Kb1
(1 + Kb2 )Ka1
(9b)
Kdapp
flux holds for all the transitions, in particular for the IF
transition (b2 [ES] − r2 [E ∗ S]), which is strictly positive
for nonzero c. In other words, the transition E → ES →
E ∗ S is more frequent than E ∗ S → ES → E whereas
E → E ∗ → E ∗ S is less frequent than E ∗ S → E ∗ → E.
With this net flow around the cycle, non-Michaelian behaviors arise. For instance, the fraction of occupation
of the enzyme by the substrate is no longer an hyperbola but becomes a nonlinear function of the substrate
(Appendix G). The enzymatic velocity takes the general
form
v=c
(11)
(where the constants A, B . . . are defined in Appendix
H). This formula can give the expected sigmoidal rate
of glucokinase for a range of parameters. To quantify
this sigmoidicity, the extent of cooperativity is classically
evaluated through the Hill coefficient nH , which cannot
exceed 2 in the present case
p
(12)
nH = 2/(1 + AD/BC)
and the enzymatic reaction rate is once again
Michaelian: v = VM [S]/(Kdapp + [S]) with Kdapp defined
above and VM = c Kb2 /(1 + Kb2 ) (Appendix E). We
verify that the nonlinear mechanism of conformational
memory does not exist in equilibrium [20].
2.2
A[S] + B[S]2
C + (A + D)[S] + B[S]2
(Appendix I). Positive cooperativity (Sigmoid) is obThe mixed IF/CS cycle in nontained
for nH > 1, that is to say when BC > AD. Interequilibrium steady state
estingly for glucokinase, a Hill coefficient of nH = 1.5 has
been measured at high ATP concentrations whereas at
low ATP concentrations (when c ≈ 0), nH = 1 [14, 16],
confirming the Michaelian activity in absence of the net
cyclical flux, that is the primary condition for obtaining kinetic cooperativity through conformational mem(10a) ory. To conceive the importance of its cyclical nature,
note for comparison that no kinetic cooperativity is possible in the linear scheme of pure CS described previously,
in which E ∗ had also the dual possibility to either relax or
rebind a substrate molecule. The net cyclical flux should
also be oriented in the direction of the IF path. As a
(10b)
matter of fact, the general form of velocity classically
admitted for glucokinase is not obtained if we artificially
force the net flux to be counterclockwise (Appendix J).
Hence, CS is clearly not sufficient in the present context and the positive net IF flux is necessary for kinetic
cooperativity to appear, unless the accepted conclusion
of conformational cooperativity is wrong for glucokinase
and that its behavior results in fact from a completely different mechanism not based on conformational changes
[26]. This remaining possibility is described below.
Equilibrium can be broken if the enzymatic rate constant
c is not negligible compared to the other rate constants.
In this case, a net circulation appears in the cycle, clockwise with respect to Fig.3 and whose value is
JN F =
cr1 b2 a1 [S]
D
with
D = (r1 + b1 )(d1 r2 + (d1 + b2 )(d2 + c))
+ (r1 (r2 + b2 ) + (d2 + c)(r1 + b2 )) a1 [S]
+ (b1 (d1 + b2 ) + r2 (d1 + b1 )) a2 [S]
+ (r2 + b2 ) a1 a2 [S]2
3
Figure 3. First-order scheme of enzyme state recycling,
mixing CS, IF and product release (rate c). NF: net flux
arising out of equilibrium (the ”turning wheel” of Wyman
[25])
Monomeric sigmoidicity without IF and CS
The above considerations clearly show that CS is insufficient to explain the sigmoidal dependence on glucose of
glucokinase through the model of kinetic cooperativity.
JN F is maximal at the substrate concentration can- Hence, it is necessary to examine if alternative models
celling the derivative of this function (Appendix F). This can give this result while allowing dominant CS. One
4
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corresponding to the IF path, opposite to the CS path
and allowing kinetic cooperativity. Hence, the claim that
CS is always sufficient seems to not apply to kinetic cooperativity, which primarily relies upon IF. This crucial
role of IF does not exclude the existence of pre-equilibria
between many enzyme isoforms. Dynamic disorder is a
widespread property of protein folding which may interfere with binding and the present report is not aimed at
minimizing the importance of CS. The categories of enzyme states defined here (E, E ∗ , ES and E ∗ S) can be
viewed as representative of many sub-isoforms without
altering the results. For example, fluctuating Michaelian
enzymes still display globally Michaelian behaviors as
long as the topology of the network and the associated
net fluxes are maintained [30]. In the enzymatic reaction studied here involving the joined CS/IF scheme, the
driver transition breaking equilibrium is the rate constant c. It has been considered completely irreversible for
simplicity, but a reversible transition unbalanced enough
to break equilibrium would have been sufficient. The
simplified one-way arrow commonly used in enzymology
without corruption of the results, is not elementary since
it covers at least two more elementary transitions: the
transformation of substrate(s) into product(s) and the
subsequent release of the product(s). The apparent irreversibility of enzymatic reactions in the usual assay conditions is not due to the fact that enzymes work in a
one-way manner with a null rate constant for the reaction reverse to the catalysis as often assumed, but simply reflects the negligible steady state product concentration. Indeed, a reaction is evolutionary selected when
its product is useful for subsequent reactions or syntheses. As a consequence, the stationary concentration of
the product is very low in the cell, thus preventing its reuptake by the enzyme. This pumping must of course be
compensated by a permanent replenishment in substrate,
reflecting the open nature of cellular systems. This situation is mimicked in vitro when measuring initial velocities, when the concentration of products in the mixture is
still negligible. In the context of the present report, the
micro-irreversibility of c has two essential virtues: (i) it
breaks the cyclic detailed balance rule and (ii) it directs
the net flow. The topology of the steady state scheme
of Fig.3 and its net IF flux, illustrate well the organizational potential of non-equilibrium mechanisms [31] and
the sentence of Prigogine ”Nature begins to see out of
equilibrium”. In the present example, a single molecule
of glucokinase, without the need for regulating its synthesis or of other biochemical circuits, can act as (i) a
sensor of glucose concentration, a role also suggested in
the pancreas, (ii) an enzyme with a conditional activity and (iii) a regulator of glucose concentration in the
blood, working in a real time manner but not by corrective feedback. These properties may explain the evolutionary conservation of the Hill coefficient of the glucokinase [16] and the importance of this enzyme in the
such model exists, this is the random binding of different substrates on an enzyme catalyzing bimolecular reactions. It could in principle apply to glucokinase which
has two substrates: glucose and ATP. The fixation of
two substrates represented in Fig.4 is theoretically sufficient to give rise to non-Michaelian rates [27] and, in
particular, to sigmoidal rates [27, 28].
Figure 4. Random building of a ternary complex containing
the enzyme and two different substrate molecules.
Though simple, this system is subtle enough: (i) The
completely hierarchical binding of the substrates, for example S1 always before S2 , gives a Michaelian reaction
rate. (ii) A completely random filling also leads to a
Michaelian behavior. (iii) But interestingly, the incomplete dominance or preference for one substrate, gives
non-Michaelian curves: either sigmoidal or bell-shaped
with a maximum, depending on the relative values of
the constants. The conditions giving these results, the
corresponding inflexion points and the apparent Hill coefficients, are detailed in [28]. For this ternary complex mechanism, conformational changes are possible but
dispensable. Similar results are indeed obtained with
rigid binding, for example in the case of asymmetrically
charged ligands [28]. This origin of cooperativity has
been ruled out for glucokinase [29], since glucose binds
before ATP to glucokinase and the binding of ATP remains Michaelian for all the doses of glucose. Moreover,
large concentrations of ATP which would alter the binding hierarchy of the substrates, do not affect the cooperativity with glucose [16].
4
Conclusions
CS is definitely insufficient to yield the kinetic cooperativity of glucokinase given that (i) Pure CS and IF modes
of binding are unable to explain the sigmoidal dependence on glucose of glucokinase, suggesting the existence
of a mixed cycle combining CS and IF. (ii) In the dual
cycle encompassing both CS and IF under the rapid preequilibrium assumption, the relative importance of CS
and IF is joined by the detailed balance to the relative
affinities of the two isoforms for the substrate. In this
condition, kinetic cooperativity is impossible. (iii) In
non-equilibrium steady state, a net cyclical flow appears,
5
bioRxiv preprint first posted online Mar. 24, 2016; doi: http://dx.doi.org/10.1101/045591. The copyright holder for this preprint (which was
not peer-reviewed) is the author/funder. All rights reserved. No reuse allowed without permission.
regulation of glycemia suggested by mutations [32]. For [11] E. Whitehead, The regulation of enzyme activity and
allosteric transition. Prog. Biophys. Mol. Biol. 21 (1970)
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321-397.
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forms of the enzyme. This difference is itself directly repossible kinetic model based on substrate-induced conforlated to the asymmetry between forward and backward
mation isomerization. Biochem. J. 102 (1967) 22C-23C.
fluxes [33].
∗
G(E S) − G(E) = −RT ln(JIF /J−IF )
[13] H. Niemeyer, M. de la Luz Cŕdenas, E. Rabajille, T.
Ureta, L. Clark-Turri, and J. Penaranda. Sigmoidal kinetics of glucokinase. Enzyme 20 (1975) 321-333.
(13)
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multimeric enzymes [31]. Multimerization can be viewed
(1976) 7-14.
as an optimized recipe in which the classical thermody[15] A.C. Storer, and A. Cornish-Bowden, Kinetic evidence
namic cost has been replaced by a structural information
for a ’mnemonical’ mechanism for rat liver glucokinase.
which had itself a past evolutionary cost [34]
Biochem. J. 165 (1977) 61-69.
[16] M.L. Cárdenas, E. Rabajille, and H. Niemeyer, Kinetic
cooperativity of glucokinase with glucose. Arch. Biol. Med.
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for phosphofructokinase. Biochem. J. 98 (1966) 278-283.
[29] A. Cornish-Bowden, and A.C. Storer, Mechanistic origin
of the sigmoidal rate behaviour of rat liver hexokinase D
(’glucokinase’) Biochem. J. 240 (1986) 293-296.
[33] D.A. Beard, and H. Qian, Relationship between thermodynamic driving force and one-way fluxes in reversible
processes. PLoS One 2 (2007) e144.
[30] W. Min, I.V. Gopich, B.P. English, S.C. Kou, X.S. Xie,
and A. Szabo, When does the Michaelis-Menten equation
hold for fluctuating enzymes? J. Phys. Chem. B 110 (2006)
20093-20097.
[34] D. Michel, Life is a self-organizing machine driven by the
informational cycle of Brillouin. Orig. Life Evol. Biosph. 43
(2013) 137-150.
[31] D. Michel, Basic statistical recipes for the emergence
of biochemical discernment. Prog. Biophys. Mol. Biol. 106
(2011) 498-516.
7
bioRxiv preprint first posted online Mar. 24, 2016; doi: http://dx.doi.org/10.1101/045591. The copyright holder for this preprint (which was
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Appendices
A
Relaxation times
IF has been differentiated from CS through the mean times of equilibrium restoration following perturbation by
changing substrate concentration. When assuming slow conformational changes and rapid ligand turnovers, as the
ligand concentration increases, the relaxation time increases in IF and decreases in CS [19]. Hence, relaxation times
have logically been used as a diagnostic to appraise the relative contributions of CS and IF and led authors to
conclude for instance that CS is predominant for glucokinase previously thought to obey IF [2]. The interpretation
of relaxation times is however more delicate in absence of the poorly justified hypothesis of time scale separation
between the transconformation and binding phenomena [6]. It could be usefully completed by complementary
kinetic studies [8].
B
One-way fluxes
The one-way fluxes passing through the CS and IF specific branches of Fig.2 are
1
1
+
b1 [E] a2 [E ∗ ][S]
−1
1
1
+
a1 [E][S] b2 [ES]
−1
JCS =
and
JIF =
=
b1 a2 [E][S]
r1 + a2 [S]
(B.1)
=
a1 b2 [E][S]
d1 + b2
(B.2)
Solving JCS = JIF gives the same critical value of [S] as the single molecule probability approach of the main text.
C
Identity of forward and backward CS and IF fluxes in equilibrium
On the one hand, the global path rates are
• kCS = b1 a2 [S]/(r1 + a2 [S])
• k−CS = d2 r1 /(r1 + a2 [S])
• kIF = b2 a1 [S]/(d1 + b2 )
• k−IF = d1 r2 /(d1 + b2 )
and on the other hand, the relative concentrations of the different forms of the enzyme are related through equilibb1 a2
[E][S]. Coupling these relationships to the global rates
rium constants in equilibrium, as for example [E ∗ S] =
r1 d2
implies the identity of the forward and backward fluxes after simplification, such as
k−CS [E ∗ S] =
D
a2 b1 [E][S]
= kCS [E]
r1 + a2 [S]
The cyclic equilibrium rule
When the enzyme-substrate mixture is in equilibrium, the detailed balance rule or generalized micro-reversibility
for cycles (Eq.(8)), also known as Wegscheider’s relation, directly ensues from the product of the rate constant
∗
ratios which are related to the energies of each species, such as b1 /r1 = e[ε(E)−ε(E )] . Hence, the circular product
of these ratios [31] is
∗
∗
∗
∗
b1 a2 r2 d1
= e[ε(E)−ε(E )]+[ε(E )−ε(E S)]+[ε(E S)−ε(ES)]+[ε(ES)−ε(E)] = e0 = 1
r1 d2 b2 a1
8
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An equivalent and even simpler way is to express the equilibrium constants as concentration ratios. Based on
the familiar relationship
∆G = −RT ln Ka1 + RT ln
[ES]
[E][S]
(D.1)
it is clear that at equilibrium, when ∆G = 0,
Ka1 =
[ES]
[E][S]
in the same manner,
[E ∗ S]
[E ∗ S]
[E ∗ ]
, Ka2 = ∗
, Kb1 =
[ES]
[E ][S]
[E]
so that after simplification, the product of the constants expressed in this form gives
Kb2 =
Ka1 Kb2 = Ka2 Kb1 =
E
[E ∗ S]
[E][S]
Enzymatic reaction rate in quasi-equilibrium
The enzymatic reaction rate proportional to the probability for a single enzyme to be in state E ∗ S
v = c P (E ∗ S)
(E.1)
and
[E ∗ S]
(E.2)
[E] +
+ [ES] + [E ∗ S]
where each concentration can be expressed as a function of [E] through equilibrium constants as shown above, so
that the fraction can finally be simplified by eliminating [E] and gives the Briggs-Haldane velocity, with VM and
KM given in the main text.
P (E ∗ S) =
F
[E ∗ ]
Substrate concentration giving the maximal net flux
JN F has the general form
α[S]
β + γ[S] + δ[S]2
where the constants are obtainable from Eq.(10). It is maximal when
p
[S] = β/δ
JN F =
giving the flux
JN F =
G
α
√
γ + 2 βδ
(F.1a)
(F.1b)
(F.1c)
Enzyme occupancy by the substrate in steady state
Y = P (ES) + P (E ∗ S) =
where A, B, C, D are the following constants:
• A = a1 r1 (b2 + r2 + d2 + c) + a2 b1 (b2 + r2 + d1 )
• B = a1 a2 (b2 + r2 )
• C = (b1 + r1 )(d1 r2 + (b2 + d1 )(d2 + c))
• D = a1 b2 (d2 + c) + a2 d1 r2
9
A[S] + B[S]2
C + (A + D)[S] + B[S]2
(G.1)
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H
General velocity formula
The general velocity formula corresponding to the enzymatic reaction of Fig.3, is
v = c P (E ∗ S) = c
A[S] + B[S]2
C + (A + D)[S] + B[S]2
(H.1)
with
• A = a1 b2 r1 + a2 b1 (d1 + b2 )
• B = a1 a2 b2
• C = (b1 + r1 )(d1 r2 + (d1 + b2 )(d2 + c))
• D = a1 (r1 r2 + (b2 + r1 )(d2 + c)) + a2 r2 (b1 + d1 )
I
Hill coefficient
The Hill coefficient is the extreme slope (minimal or maximal) of the so-called Hill plot drawn in Logit coordinates.
Starting from
v
A[S] + B[S]2
(I.1a)
=
VM
C + (A + D)[S] + B[S]2
A[S] + B[S]2
v
=
VM − v
C + D[S]
(I.1b)
the Hill function H = ln(v/(VM − v)) can be examined directly in Logit coordinates. If writing x = ln[S], its first
and second derivatives are
A
C
H 0 (x) = 1 −
+
(I.1c)
A + B ex
C + D ex
H 00 (x) =
A
A2
C
C2
−
−
+
A + B ex
(A + B ex )2
C + D ex
(C + Dex )2
The substrate concentration giving the extreme slope is obtained by solving H 00 (x) = 0
r
AC
[S] =
BD
(I.1d)
(I.2)
giving, when reintroduced into H 0 , a Hill coefficient of
2
r
nH =
1+
AD
BC
(I.3)
nH has a maximum value of 2 and is higher than 1 (sigmoidal dependance on substrate) for constant values
such that BC > AD.
J
Comparative results of IF and CS net fluxes on enzymatic velocity
The net flux arising in the mechanism of kinetic cooperativity is clockwise with respect to Fig.3. Let us push the
asymmetry of the flow to its maximum. To this end, E ∗ is supposed to be a high energy species which rarely
appears spontaneously from E (b1 = 0), but is induced by the substrate. In turn, E ∗ S rarely reverses to ES owing
to the stabilizing effect of the substrate (r2 = 0). In the same way, the tightly bound substrate rarely leaves the
complex E ∗ S unless it is converted into a product of different shape/charge (d2 = 0). The absence of spontaneous
substrate dissociation is logical when the binding site closes after wrapping around the substrate. In these simplified
but acceptable conditions, the reaction rate reads
v=
ca1 b2 (r1 [S] + a2 [S]2 )
cr1 (d1 + b2 ) + (b2 c + b2 r1 + cr1 )a1 [S] + b2 a1 a2 [S]2
10
(J.1)
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which can be fitted to the observed sigmoidal glucose concentration-dependent rate of glucokinase. For comparison, if we artificially force the net flux to be clockwise, in the orientation of the CS path, by cancelling r1 and b2 ,
we would obtain a bell-shaped velocity curve vanishing for high substrate concentration, in contradiction with all
the observations.
v=
K
cd1 b1 a2 [S]
d1 b1 (r2 + c) + (d1 r2 + b1 r2 + d1 b1 )a2 [S] + r2 a1 a2 [S]2
(J.2)
The flux-energy relationship
Eq.(D.1) can be rearranged as
a1 [E][S]
d1 [ES]
(K.1)
∆G = −RT ln(J+ /J− )
(K.2)
∆G = −RT ln
that is simply
11